Climbing the experience rankings is a way to show that you have mastered SMR. But it's not just for show, because as you increase in level, you become more skilled at piloting your ship. In combat, you are better able to dodge incoming weapon fire, and you hit your target more often. Out of combat, you are less likely to misjump when using your Jump Drive (if your ship is equipped with one), and you will be detected by fewer enemy ships when your Cloaking Device is activated (again, if your ship has one).
Experience is gained by trading skillfully at ports, building on planets, and participating in combat. You can lose experience too. This can happen when you dump cargo from your ship, or if your ship gets destroyed.
List of Experience Levels